1. Field of the Invention
This invention relates to the field of compound winding apparatus and to counterbalance systems for window sashes and the like, and more particularly, to a compound counterbalance system incorporating a compound winding apparatus with a zero torque spiral configuration.
2. Prior Art
Counterbalance systems for window sashes and the like have been known for some time, particularly in conjunction with double hung window frames. The original counterbalance systems for such windows utilized sash weights hung by cables and pulleys, and running in large cavities to one or both sides of the window frames. Such counterbalance systems made such windows practical, but at the same time, were extremely energy inefficient. The large cavities in which the pulleys were disposed provided effective conduits for drafts running along and through the windows and the walls in which the windows were mounted. Such counterbalance systems did have one advantage, namely that the pull exerted by the sash weight was constant throughout the reciprocating range of movement of the window sashes.
A number of developments have impacted on the design of windows and counterbalance systems for windows, resulting in the inevitable obsolescence of the sash weight and pulley system. Technology has been developed to manufacture windows much less expensively off-site, in fully functioning assemblies. Such assemblies incorporate within their own structure the necessary counterbalance system. Cavities for sash weights to counterbalance windows are no longer even designed or otherwise provided for today. Accordingly, spring balances were incorporated into such manufactured window assemblies. Spring balances are advantageous in that their tension can be easily adjusted, whereas changing sash weights is a major undertaking. The ability to provide adjustable tension has proven especially important today, as energy conservation requirements demand that windows be very tightly sealed against drafts and that they provide substantial insulation. It is not generally appreciated by consumers at large that highly efficient weather stripping and weather seals exert large amounts of sliding friction, making windows and other systems incorporating such seals much more difficult to operate. Adjustably tensioned springs permit manufacturers to compensate for the additional tension which is necessary for a well-sealed and well-insulated window. Insulated windows require additional panes of glazing, increasing the weight of the sash. However, counterbalance systems incorporating springs have simply never worked as well as did sash weights, when windows were substantially unsealed, because such springs have inherent unconstant spring rates or gradients, that is, the amount of tension exerted by the spring changes according to the extent to which the spring is extended or relaxed.
In today's marketplace, window manufacturers are faced with two critical goals, namely achieving thermal efficiency (i.e., insulation and very low air infiltration), and at the same time, achieving low operating forces. There is a direct conflict in these two goals, as tightly-fitted, heavy window assemblies inherently generate higher operating forces due to friction and gravity. Today's counterbalance systems are simply not responsive to today's needs.
As might be expected, the prior art is replete with counterbalance systems for window sashes and the like, in what seems to be, at least initially, the widest variety of mechanical systems. A number of patent references disclose the use of spiral drums to compensate for tension changes in a spring, but in each instance, the spiral drum appears to have been mounted coaxially with and axially driven by the spring, and both the spring and the spiral drum have been disposed in a cavity above or below the window. Moreover, none of these patent references has used the spiral drums in further combination with a pulley system or other means for imparting a mechanical advantage, which in turn increases the effective range of movement of the biasing means. The following United States patents are representative of such teachings: 97,263; 1,669,990; 2,010,214; 2,453,424; 3,095,922; 3,615,065; and, 4,012,008.
Patent references have also disclosed windows utilizing side mounted springs and pulley systems, most of the pulley systems being arranged in a block and tackle arrangement to achieve a mechanical advantage. However, the systems disclosed in these references require all available space, and none incorporates a means compensating for changes in spring tension. United States patents representative of such teachings include: 2,262,990; 2,952,884; 3,046,618; 3,055,044; 4,078,336; and, 4,238,907.
Certain references have also provided alternative solutions to compensation of variable spring rates in tensioning means, other than spiral drums and in other contexts. In U.S. Pat. No. 4,389,228 the sheaves of the pulleys are so close together that only one diameter of rope or cable can fit there between. The effective diameter of the pulley therefore changes with each rotation as more of the cable is wound onto, or paid out from the drum. Other solutions are disclosed in the following U.S. Pat. Nos.: 2,774,119 and 3,335,455.
3. Related Applications
This application represents further development of the invention disclosed in commonly owned U.S. application Ser. No. 892,704, now U.S. Pat. No. 4,760,622.
The invention of the commonly owned application was embodied in a new compound winding apparatus and counterbalance system for window sashes and the like. A compound counterbalance system according to that invention is the first such system sufficiently compact and sufficiently efficient to interconnect and utilize: (1) an axially expansible and contractable biasing means; (2) a constant rate of movement system providing a mechanical advantage, and reduction in space requirements; and, (3) a variable rate of movement system to automatically compensate for inherent variability in the tension of the biasing means. Moreover, the biasing means itself is adjustable. Finally, compound counterbalance systems according to that invention are easily incorporated into off-site manufactured assemblies.
The term mechanical advantage, as used in the constant rate of movement system, requires some clarification to be meaningful in context. There is a "cost" for every mechanical advantage. In the context of pulleys, as used in block and tackle assemblies, one can achieve significant mechanical advantage in raising a heavy load, but the load moves at a speed which is inversely proportional to the ratio of mechanical advantage, that is, much slower. The distance through which the load moves is also much less than the supporting cable at its driven end. In the context of gear systems, the "cost" is a rotational speed reduction. Where speed is more important than power, a mechanical disadvantage is preferred, as in an automobile's overdrive transmission. In the context of levers, a longer moment arm for the driven end of a lever will move a heavier load, but through a shorter distance, relative to the driven end. For this invention, a window sash or the like must move further than the expansion space available for, as an example, an axially extensible spring; considerably further.
The various mechanical advantage systems utilized in that invention enable maximum range of sash movement and, at the same time, minimum range of movement for expansion for appropriate parts of the biasing means. Nevertheless, the various embodiments maintain a mechanical advantage in stressing or extending the biasing means. Compound counterbalance systems according to that invention successfully exploit the "cost" of mechanical advantage systems without, in fact, sacrificing all of the benefits. Moreover, the variable rate of movement system, which is embodied in a compound winding apparatus and compensates for variability in the tension of the biasing means, can be embodied in small dimensions which further reduce space requirements for window sashes and in other applications.
The system described in U.S. patent application Ser. No. 892,704 represented a radical departure from the prior art, and in that regard, provides many, very significant advantages. However, under certain load and operating conditions, the system is subject to problems. In a mechanical sense, these problems flow from the direct connection of the variable rate spring to the variable diameter portion of the pulley; the fixed diameter or drum portion of the pulley being connected to the sash, through the constant rate of movement system.
4. Theoretical Considerations
In the prior commonly owned patent the sash force, which is constant, is applied to a constant radius drum while the linearly changing spring force is applied to a spiral of constantly changing radius. The arrangement is illustrated schematically in FIG. 2a. As the force due to the spring increases, the moment on the spiral remains constant since the radius of the spiral decreases. This relationship can be expressed mathematically, as shown in Equation 1. The left side of the equation is representative of the sash/drum side of the system where "F" is the constant force due to the sash (and which may be deemed to include the constant effect of the constant rate of movement system) and "r.sub.d " is the constant drum radius. The linearly changing spring force and changing spiral radius are represented by "x" and "y", respectively, on the right side of the equation. EQU F.multidot.r.sub.d =x.multidot.y Equation 1
Since "F" and "r.sub.d " are constants, their product is a constant throughout the operation of the sash and the left side of the equation can be represented as the single constant, "d".
The force due to the spring is a linearly changing variable and so is raised to an exponent of the first power or "x.sup.1 ". The relationship between the spiral radius and its rotational position is given by Equation 2. ##EQU1##
where:
o=rotational position of spiral in radians; PA1 s=sash force/spring constant; PA1 r=radial position of spiral; and, PA1 C=constant of integration. PA1 d=sash/drum constant; PA1 x=spring force; and, PA1 y=spiral radius. PA1 d=sash/drum constant; and, PA1 z.sup.3 =spiral/spring relationship. PA1 independent constant=sash/drum constant PA1 dependent variable=spiral/spring relationship PA1 F=sash force; PA1 y=spiral radius; PA1 x=spring force; and, PA1 rd=drum radius. PA1 F=sash force; PA1 r(.theta.)=spiral radius as a function of rotational position; PA1 F.sub.s =spring force; and, PA1 r.sub.d =drum radius. PA1 L=length of spring extension per revolution; and, PA1 r.sub.d =drum radius. PA1 F.sub.s =spring force per revolution; PA1 K=spring constant; and, PA1 r.sub.d =drum radius. PA1 F.sub.s =spring force; PA1 r(.theta.)=spiral radius as a function of rotational position; PA1 K=spring constant; PA1 r.sub.d =drum radius; and, PA1 .theta.=rotational position of spiral. PA1 r(.theta.)=spiral radius as a function of rotational position; PA1 K=spring constant; PA1 r.sub.d =drum radius; and, PA1 .theta.=rotational position of spiral. PA1 F=sash force; PA1 y=spiral radius; PA1 x=spring force; and, PA1 r.sub.d =drum radius.
As is seen, there is a squared relationship between the radial and rotational position of the spiral such that the changing spiral radius, "y", of equation 1 is raised to the second power and is represented as "y.sup.2 ". Replacing the above parameters into Equation 1 results in Equation 3. EQU d=x.sup.1 y.sup.2 Equation 3
where:
The terms "x" and "y" are in a dependent linear relationship since a change in "x" necessitates a proportional change in "y". They may therefore be multiplied for illustration and be represented as shown in Equation 4. EQU d=z.sup.3 Equation 4
where:
This illustrates the cubic relationship between the sash/drum side of the system and the spiral/spring side of the system.
The system may also be analyzed as the combination of independent and dependent variables. Since there is no change in the force of the sash or in the radius of the drum, these variables are independent of any other factors at any given time and are, therefore, constants. Conversely, both the spring force and the spiral position are dependent upon the location of the sash and upon each other at all times. The previous example, in terms of Equation 1, would be expressed as Equation 5. EQU independent.multidot.independent=dependent.multidot.dependent Equation 5
Equation 5 is "unbalanced", as between independent and dependent variables. The dependent variables are affected not only by external factors but also by each other, which leads to their cubic relationship. Relating the multiplication of independent and dependent variables in terms of Equation 4 would be expressed as Equation 6: EQU independent constant=(dependent variable).sup.3 Equation 6
where:
The relationship, which can be thought of as "unbalanced", creates problems in a number of areas. The first of these problems is that of force multiplication over the spiral. Since the cord from the sash/reduction block must wrap on the drum, the drum must be of a substantial diameter to wrap this cord in only a few turns. For example, in a system designed for a sash with a 24" travel on a 4:1 reduction block, a drum diameter of 0.955" is necessary to keep total rotation of the spiral/drum under 2 revolutions. If this diameter were to be decreased, the number of revolutions of the spiral/drum would increase and the spiral would be excessively long, which leads to space restrictions within the window frame. In the past, an inner spiral diameter of 0.50" has been found to yield a spiral of outer diameter of 1.20". This size is acceptable within the confines of the frame, however, it leads to a drum to inner diameter ratio of approximately 2:1. The weight of the sash is initially increased four-fold as it passes through the reduction block and the total force increase through the system is 4 multiplied by 2, or 8:1. Since the drum diameter cannot be reduced significantly, the inner diameter of the spiral must be increased to reduce the total force multiplication. This leads to the second problem.
There is a great deal of rotational travel in the spiral for very small changes in "r" near the inner diameter. As the radius increases, the effect decreases, as does the rotational travel. At larger diameters, there is very little rotational change between large changes in radius. In order to properly design a spiral/drum system, the spiral must be designed to rotate through the same number of revolutions as is necessary for the drum to wrap the required sash cord. Because of the large rotational changes at the smaller radius, most of the spiral length is concentrated at a small mean radius. If even a small increase is made to the inner radius, a substantial increase must be made to the outer radius to compensate for the loss in rotational travel at the smaller radius. Moreover, the force ratio between the inner and outer radius of the spiral increases dramatically. For a 100 pound sash, for example, the force ratio will cause a marked decrease in the life of any connecting cord, as it is cycled back and forth between approximately 100 pounds and 650 pounds of force.
A third problem is that of force location. When the sash is lowered, the spiral rotates and extends the spring. As the spring extends, its force increases and the spiral radius decreases. When the sash is at the bottom of its travel, the spring is extended to its fullest length and therefore exerts the greatest force in the system. This high force is applied to the smallest radius of the entire spiral and creates very high compressive stresses in the spiral, which can cause premature failure of the spiral at this point. An alternative means for counterbalancing a sash, or the like, was developed to overcome those problems of the initial breakthrough. The underlying concept of this invention is to separate the independent and dependent variables of Equation 5, so as to be in a "balanced" relationship. Accordingly, the constant force of the sash was applied to the radially changing spiral and the force of the linearly changing spring was applied to the constant radius drum, so that Equation 1 is rewritten as Equation 7. EQU F.multidot.y=x.multidot.r.sub.d Equation 7
wherein:
In this arrangement, which is illustrated schematically in FIG. 1a, the moments on both sides of the system are changing whenever the sash is in motion, but they are compensating each other and maintaining a net zero torque on the compound aspect of the system as a whole. The system has accordingly been designated the Zero Torque Spiral System (ZTS). In order to accomplish this net zero torque, the equation which determines the spiral plot had to be developed. The moment on the spiral side of the system is equal to the force from the sash multiplied by the radius of the spiral, as a function of its rotational position. The opposing moment is caused by the linearly changing spring force on the drum. Equating these factors to maintain a zero torque at any rotational position yields Equation 8. EQU F.multidot.r(.theta.)=F.sub.s .multidot.r.sub.d Equation 8
wherein:
Since the spring force is a function of the spring's linear extension due to the rotation of the drum, this extension is a function of the drum radius. For each full revolution of the drum the spring will extend as shown in Equation 9. EQU L=2.pi.r.sub.d Equation 9
wherein:
The force due to this extension is equal to the product of the spring constant and the spring's extension as shown in Equation 10. EQU F.sub.s =K.multidot.2.pi.r.sub.d Equation 10
wherein:
This is the spring force achieved after one revolution of the drum. Since there are 2.pi. radians in each revolution, the spring force per radian can be determined and, if there are .theta. radians in the rotation of the drum through any fractional rotation, the spring force at any rotational point is shown in Equation 11. EQU F.sub.s =K.theta.r.sub.d lbs. Equation 11
wherein:
Substituting this into Equation 8 and solving for .theta. results in Equation 12, which defines the Zero Torque Spiral. ##EQU2## where: F=sash force;
At any point on the spiral and at any position of the sash, the sash force, spring constant and drum radius will be constant at all times. Accordingly, these terms may be factored out of formulating the spiral radius, yielding Equation 13. EQU .theta.=a r(.theta.) Equation 13
wherein: ##EQU3## Equation 13 proves to be the general form for an Archimede's spiral and defines a linear relationship between the rotational and radial positions of the spiral.
Since "r" is now a linear function of the rotational position of the spiral and the spring force is a linear function of the spring extension, both "x" and "y" of Equation 7 are raised to the first power as shown in Equation 14. EQU F.multidot.y.sup.1 =x.sup.1 .multidot.r.sub.d Equation 14
wherein:
This can be illustrated in the alternative form of independent and dependent variables, as shown in Equation 15. EQU independent.multidot.dependent.sup.1 =dependent.sup.1 .multidot.independent Equation 15
By separating the dependent variables, the general moment equation has been reduced to a direct linear relationship between both sides of the equation. For a unit change in the "y" parameter, there will be an equivalent unit change in the "x" parameter.
The theoretical analysis is well summarized by typical graphs of rotational versus radial data. Data for the unbalanced relationship is shown in FIGS. 2b and 2c. FIG. 2b illustrates the exponential relationship between the spiral radius and its corresponding rotation. FIG. 2c shows a very steep slope in this relationship for small radii and a slope which approaches zero very rapidly as the radius is increased. This accounts for very little change in spiral rotation even through large ranges of radius. FIGS. 2b and 2c graphically illustrate the concentration of the exponential spiral in an area of small radius with a considerable waste of space in the large radii. Data for the direct linear relationship of the Zero Torque Spiral System is shown in FIGS. 1b and 1c. As is shown in FIG. 1b, there is a direct and constant increase of rotation by 10 radians for every 0.50 inch increase in spiral radius. Another difference is the length of the Zero Torque Spiral over the exponential spiral. Traveling between the same radii, the exponential spiral has a length of 8.08 inches while the Zero Torque Spiral has a length of 21.86 inches. For a given 5:1 reduction ratio, this allows the Zero Torque Spiral a sash travel capability of 109.3 inches versus 48.0 inches for the exponential spiral. In this configuration, the maximum force in the exponential spiral is 175 lbs. while the maximum force in the Zero Torque Spiral is 262.5 lbs.; however, equating maximum force yields a Zero Torque Spiral of outer radius equal to 1.0 inch with a spiral length of 9.38 inches. This allows a sash travel of 46.9 inches which is 1.1 inches shorter than the larger exponential spiral. The Zero Torque Spiral has its maximum force on its 1.0 inch diameter drum in either situation, as opposed to the exponential spiral which has 175 lbs. on a 0.50 inch diameter inner radius.
A number of problems are solved with the development of the Zero Torque Spiral System. The first and most important of these is the relocation of the maximum force in the system. At all points on the spiral, from inner to outer radius, the force is constant, and is the force due to the sash through the reduction block (constant rate of movement system). The point at which the higher forces due to the extension of the spring are located is on the constant radius drum. The drum is of a radius which is consistently larger than the inner radius of the spiral and, for design considerations, is as large as possible. In addition, the force on the drum is distributed over a larger, more uniform cross-sectional area rather than on the single groove of the inner radius of the spiral. This leads to a greatly reduced potential for material failure in this area due to compressive loading.
The second improvement is that of the linear relationship of the spiral radius. Since the spiral is linear in configuration, the greater portion of the spiral travel is in a larger mean radius and the relationship between the inner and outer radii is linear, so that a change in one radius will cause a comparable change in the other. Because the force concentration on the spiral is no longer a consideration, it is not necessary to increase the inner radius to compensate for this; however, certain connecting cords and cables have minimum bend radii and if it were necessary to increase the inner radius, it would not drastically affect the outer radius.
The forces in the ZTS are, in general, the same to somewhat less then those in the unbalanced system, depending upon the design of the system. Importantly, however, the forces in the zero torque spiral system are applied at locations where they are much more tolerable and easier to deal with.